I'm very curious how you're going to do that. I wonder if you'll define e with a series and then prove d(e^x)/dx = e^x or if you'll define e to be the number such that this property holds.
It is amazing, but the problem is, I don't think students learning calculus will understand/appreciate it unless they already know calculus &/ already have a solid grasp on mathematical thinking.
@@yaakovasternberg6295 I disagree. These videos were what I used to learn these concepts before I learned further with Khan Academy videos. Starting with visual intuitions made the math easy to memorize as it made sense.
This series, Essence of Calculus, is like the Feynman lectures on Physics. It's a masterpiece. It is simply the way the subject should be taught. Congrats, I love this series
Stephen Sinnemann. It is amazing to learn calculus with real interpretation and understanding, and not just repeating formulas... I am very thankfull for 3blue 1 brown, because his explanaitiona are extramly clear and take you back to basic thinking!
This whole playlist is gold, you deserves a statue. Also, I was wondering which program do you use for animations, since they do math stuff that's probably not easy to achieve with a normal animation program. Googled it, found that you wrote a python library yourself to achieve this, the respect bar goes higher and higher.
Yes this is an impressive and very useful work. As a teacher I share it with my students. I find that I am jealous of them since I didn't have wonderful resources like this when I was learning Calculus.
Please don't stop being. I'm in university at the moment and even though I thought I had a good grasp of mathematics because of good grades, nothing excites me more than seeing something explained properly, seeing the layers of abstraction erode away and leave behind that one little piece of truth. You are hands down the best educator on RU-vid right now, Brady, CGP Grey, Crash Course, VSauce, they all wish they could teach like you can.
Wow. It's incredible. After 5 semesters of studying applied math at university I am very familiar with all those rules. But never in my education have the concepts been explained that well and rigorously. I thought going all the way down to the fine details and reasons would be boring but it's actually so much more interesting than the "standard" way to learn calculus.
This is hands down the highest quality content on calculus I've found. I sent this to my old calc professor and I really think he'll enjoy it! Can't wait to see these every day when I get home from class. Keep up the amazing work!
Thank you for this entire series. As a retired engineer I really enjoy reviewing the 'how' and 'why' of so many memorized principles used in my profession. Keep up the great work!!!
I like to call the chain rule "The Russian Doll rule". I find that gives my students a better sense of how to "unpack" the functions when differentiating. You can takeout the inner doll while it's still inside the outer doll. This is a great series, by the way. Looking forward to tomorrow's video!
@@rodolphov.santoro8829 The best example for recursion is folder in folder in folder. Ask students to write a function to parse path name (only introduce recursion after they know how to deal with strings), and they'll get the point. Fibonacci is the worst example, because you can do it with loop and easier. Don't use that, even if all textbooks do that. Your students will think recursion is just another useless pedantic nonsense.
@@y.z.6517 To complete beginners folder inside a folder is already a new concept.Though i agree to cs students it's a good idea, it's not to every public. Some people like examples with real life objects, and it's specially usefull to someone who barely uses their computers.
@@rodolphov.santoro8829 "To complete beginners folder inside a folder is already a new concept." Technically true, but anyone who can learn coding already has lots of experiences with using computer(s). I agree that russian doll is a good example, but it is non trivial to translate that to codes. You need to know how to do animation. Then, you need to know relatively advanced maths like matrix and transformation. In practice, either a student gets it as a metaphor, or he is confused.
Wish more math professors told their students why these things work and what they derive rather than reading straight from the book and writing endless examples on the board. Thank you for this amazing video, your the math professor I never had.
**TIMESTAMPS TABLE** 0:05 Initial quotation (chain rule like onion) 0:15 Last videos were about simple functions 0:30 What about derivatives of more complex functions? 0:50 There are 3 ways to combine functions: sum, multiplication and composition 1:00 Subtracting and dividing are special cases 1:50 Derivative of the sum 2:15 Example sin(x) * x^2 3:00 df is clearly the sum of dg and dh 4:00 Performing the division by dx we get to the formula 4:15 Derivative of products 4:20 Visualization of a product as an area 4:47 Each side depends on a function 5:30 Now let's analyse how a change of dx causes a change of area (df) 5:40 df is the sum of 2 rectangles (+ an infinitesimal of second order) 6:30 Remember that dx -> 0 6:45 Working out specific example 7:20 General 7:30 Mnemonic Left d-right Right d-left 7:55 May be strange if arbitrary but now you know the rectangles areas 8:20 D k*f(x) = k * D f(x) 8:45 Derivative of function composition 8:50 Example sin(x^2) 9:07 Another visualization: 3 number lines, x, x^2, sin(x^2) 9:45 Let's analyse a tiny nudge dx 12:00 Derivative of the outside with respect to the inside * Derivative of the inside 12:50 This is called the "chain rule" 13:05 What df / dh means 14:10 The simplification of dh represents a fundamental concept 14:40 Now you have the 3 basic tools 15:00 Please do practice calculations 15:40 Patreon supporters 15:48 Ads
+ErikCR Added ads timestamp, I do not think it is possible to be precise to the second with these kind of time-stamps (when do you link? When he says the word or when he starts the phrase or when he start the "paragraph" or when the image is displayed, or when the formula is displayed?), and even if it was I do not think it would be so important Also there is a trade-off between practicality of time needed and precision, like this it takes me 30 minutes, second-precision possibly much more
And here I thought Brilliant gives you a good insight... I simply cannot express with words how useful this series is. Simply the best. I wish there were textbooks written like this.
I've been trying to learn calculus and specifically the chain rule for YEARS, and haven't had the motivation and time to do it. Today I finally understand it, in no small part thanks to these videos which made these concepts intuitive, elegant and exciting. Thank you so much!
Just sitting down and watching each of these videos has taught me more about Calculus than all of my years of college calc. I've lamented the fact that for me Calculus has been little more than a series of steps I go through to solve a problem rather than a process that I use to think through and understand what is what is actually going on. This video series in a single viewing has helped me more than anything else to really understand Calculus. I really look forward to this series moving on to discuss Integrals. Thank you so very much.
The product rule of derivative is so enlightening. Never knew, neither imagined! This is not just calculus. This is more like a realization through some philosophy! Pranamam GURU (Namaskara in a deepest, revered sense)
Grant, I am 12 and I have always been fascinated by math. Other than my dad, I have not found someone who teaches math well. I love you videos! Thank you for all you have done!
Knowing how math works is so much better than learning formulae in class. Not just because it gives you more knowledge/perspective but also from an exam perspective where you can come up with the formula by yourself because you know why/how it exists. Thank you so much for making this amazing series!
I'm about to finish up my bachelors in mathematics, and it's so crazy to see these things that I've known and used for years as just a formula plug in to be visualized and fully explained. Thank you so much for this video series, reminds me of why I fell in love with math in the first place :)
Was lucky enough to have a teacher in high school who explain the chain rule and other calculus topics the way they are here. Not as visually appealing, but with the same inquisitive approach to learning. It makes a whole lot of difference on having kids love the course and not dread it.
This is something that high school students need to watch before they enroll into their university, especially when you are in math-related course. There is nothing more scarier than not knowing the basic essence, or mechanics of what you are doing in uni. Here I am uni student majored in economics, learning calculus AGAIN in my holiday
I can't tell you enough how much I love this series, actually understanding what is taking for granted in math classes feels amazing and it has helped me tackle much harder problems (e.g. in math competition). You are basically my main educator when it comes to math and it is really helpful. Love what you are doing, thank you so much!
its been six years and i think of this series every single day. perhaps one of the best to explain the application and deeper understanding of calculus
Just realized from this video that perhaps the most powerful part of this series is that it connects fundamental concepts that you learn before calculus (algebra and geometry) directly to calculus. I had always thought and been taught that calculus is this crazy new way of doing math. It's not, it's just the same way of doing math with a few nice conceptual tricks and nuances.
exactly. It's wierd This "complex" idea of small changes and dt's and dx's and those rules. And yet, this can be solved algebraically that's why when d(sin(h))/dh = cos(h) h = x^2 dh/dx = 2x dh = 2x•dx dy/(2x•dx) = cos(x^2) dy/dx = (2x)cos(x^2) litterly subbing in and moving and we get the derivative
This channel has made me fall in love with Math and changed the way I used to look at things, very much appreciated and thank you for taking so much effort to create such a discriptive videos. I have a mind of a Mathematician, keep on the work of teaches others.
I know @3Blue1Brown is probably not going to read it, but I just want to thank you for everything you have created. You are a totally different vision of maths, capable to make anybody (even if they say "they don't like maths) being surprised and impressed of how 'magic' they can be. Those demonstrations, as well as the graphs and the fantastic explanations, turn you into like "woah, I think I'm quitting life and I'm going to do maths instead". On the other hand, you have created a nice, respectful community arguing about maths in such an educated way. Your explanations are so simple that practically anybody can understand them. Therefore, everybody can contribute and even somebody who is 15 can point out something a 50-year-old teacher had never thought of. So yes, thank you very much, and can't wait for another video.
I couldn't understand the derivative of the quotient of two functions. I asked for help to someone who ended the university successfully. That person spat more formulas that neither of us could understand. Now I realize how important your work is for someone looking to learn calculus, not only solve calculus problems. Thank you from Perú
You sir make me appreciate calculus in a way I never thought possible. series like this are the reason why I study subjects just for the fun of learning, not for taking a test
In calculus last year before I had heard of this channel I was told to prove that the product rule extends for f(x) * g(x) * h(x) and the algebraic proof was awful. I now tutor and after having seen this video I realized that this visualization makes that proof so much easier and so much more satisfying when you expand to 3 dimensions instead of 2.
There is a product rule with the number of functions being whatever You want. If You have f1*f2*f3*...fn, then the derivative equals: f1'*f2*f3*...*fn+ f1*f2'*f3*...*fn + f1*f2*f3'*...*fn +...+ f1*f2*f3*...*fn'. Or, the derivative of the first times all the other things plus the derivative of the secund plus all the other... It's easy to prove with Logarithmic differentiation (just take natural log of the whole function and take the derivative). :^)
Reading the comments here, I'm happy that at least I found this series before any formal calculus education. We will start basic calculus after some 6 months.
Grant, once again, well done. I personally learned the chain rule as looking at gears and how the change of one gear changed the next which changed the next, but I also really liked your explanation. keep it up!
I love that you've turned multiplication into a great tool! Back when I was taking these calculus classes and struggling to "get" the math, I felt like my inability to visualize anything other than multiplication meant I didn't understand any of it. Whereas in this video, you demonstrate the flip side: leverage whatever you've got to get yourself further.
I was initially going to study maths at uni, but I switched to Eng lit because my heart wasn't in maths anymore. Now that I'm finishing a very fulfilling Eng lit degree, I have no regrets, but I do sometimes wonder what maths would've been like. These videos make me feel like I'm doing a little bit of uni maths.
HarryIsTheGamingGeek Besides teaching, there's various positions in e.g. industry and business... ... And ALSO increasing the human store of knowlege, enriching the human experience. Money allows you to live, but science and art allow you to enjoy it. It's not just about jobs.
...................... Leibniz Notation now suddenly makes a whole heck of a lot of sense thanks to this video. I think when I go back to university, from now on when I work out derivatives I'm going to include that little dx or d[whatever] at the end to help me remember what's actually going on. And this also makes the derivative identity d/dx cos(x)sin(x) = cos²(x)+sin²(x). Trig identities have always been my Achlles' heel ever since I took precalc in high school, which I owe to a terrible precalc teacher who had an "I'm retiring" no-care attitude and an even worse textbook that only covered bare basics and then threw ugly curveballs in the problem set. A video series about how those work would be AMAZING and SUPER helpful to someone like me.
The channel MindYourDecisions has a video showing how the quadratic formula can be derived geometrically. After seeing that, I was really interested in how it could be applied to other areas of math. Needless to say, your channel is the absolute best at this. I work as a financial auditor, so I deal with math everyday, but no more complex than basic math usually, but this channel and a few others really make me want to get into deeper math concepts. Thanks for all the hard work you must put into these, they are extremely captivating!
I am very grateful for this series I was a commerce student until I decided to pick up maths again as my major. The journey is difficult but these videos help a lot..... You have made me fall in love with the subject.
These videos remain timeless, and they are so apreciated. Thank you. Not only to you explain the topics in an easily digestible way, but you also teach math as something to marvel at and something to love. You make math fun!
I love you 3Blue1Brown. You really make Math look beautiful and perfect. You introduce it to me magically again, after I had to see Math being ruined by my highschool teachers' teaching. Thank you for everything!!
I just ran across these videos. I haven't had time to watch them yet, but I can see what the approach is which is exactly the kind of thing I like. Thanks!
These are truly beautiful videos. Thank you so much. I find your explanation style crystal clear - the animations are insightful, elegant, and efficient. It'd be difficult to overstate how much of a gift you are to people who are seeking strong intuition in math. Continue, continue, continue making these please.
Every video in this series blows my mind. I wish there had been time to go over this intuition when I learned calculus. I knew calculus, but I didn't *understand* calculus, until now.
Exceptionally exceptional clarity in concepts. If wish I had found you in 2000 . All higher engineering , science topics use to go over the head due to poor grasp of math topics. U r doing exceptional service to academics as well as to future advancement of science and technology
My Calculus teacher Dr. Novák-Gselmann Eszter always told me how beautiful calculus is, and even though i always understood calculus and what she was teaching, it's only now, because of this video and the visualizations, that i realize how beautiful it really is, and i am really thankful for that, great video!
Fortunately I got to know about you through "DOS",and now I'm marvelled with your thoughts , explanation , visualisation , narration and everything else.......
This is like everything any physics teacher would teach, but exactly what my math teachers demanded we *not* do (or, at least, take seriously). For instance, "cancelling" the 'dh's in the chain rule. I'm really not sure what the value of that admonishment was. Also, learning to think along the lines of d(sin(t)) = cos(t) dt is very useful. At the very least, it's notationally useful, but I think more so. Thanks!
The cautions are valid for when we get rigorous these hand waivy differentials are harder to justify. The the hand waivy intuition is still useful, and no disrespect meant by the term, any graphical introduction is hand waivy, but it helps us learn and should be included.
After a while of rewinding, taking notes and slowly progressing through the composite function portion of this video, I slowly started to realize that I can visualize these functions as nested functions in a computer program. Where the inner function evaluates an input and the outer function evaluates whatever the inner function returns. For whatever reason, this helped me grasp this concept just a little better even though I'm pretty sure this will take a long time for me to actually understand the mathematics. Thanks for this amazing series, it is invaluable.
this channel deserves more subscribers, more followers, more people should know this channel exist. This is my goal for 2020. to tell people about this channel. i am beyond grateful for this channel and this channel proves that math is fun when you understand it.
